Let us consider the problem of two back-to-back photons in a
total spin zero state.Measurements of
this system are sensitive to the basic rules of QM. See, for example, the work
ofJ.G. Rarity for a more detailed
discussion of actual experiments.Other
topics that are often connected in these discussions include, Bell’s Theorem
and EPR.

In any review of Quantum Mechanics the questions of
interpretation arises.There are two
types of questions:

How do
we interpret and understand the QM rules?

Are
these interpretations reasonable?

There are many articles written on the subject of QM. This
short review will list a few points that help address the questions.

Some important points:

Complete
orthonormal basis.

States will be written as sums over basis vectors.

Only
amplitudes of the same basis vector interfere.

Consider a two state system with basis|1>, |2>

These states will interfere as follows:

The probability of finding the final state in the condition
|1> is

Two quantum states will not interfere unless they have
common amplitudes for particular basis vectors. For the case that a photon is
emitted in the two slit descriptions of Feynman, the basis might be:

The states cannot interfere with
the state because they are
orthonormal. The basis states above need to be developed as a two state basis
(composite system) as treated below.

Two
systems that interact or that make up a composite quantum state are
treated as the product of the separate states as follows:

System 1 à basis |a>

System 2 à basis |ά>

Composite System à
|a>|ά>

As you build states from the above
basis:

One finds that in general the
final composite systems cannot be decomposed into a product of the form:

The two systems can become entangled.

The often treated two particle
system is the two photons emitted back to back in a total spin zero state.

When the photons are separated by
a significant distance the entanglement means that an interaction of photon 1
with another system may be dependent on photon 1 polarization.

Remember that a photon can be represented just like the
Electric field vector for a traveling E&M wave. One can break the field
into two components perpendicular to the motion then specify the x and y
components. One can also use a complex coefficient, which rotates the field
around the direction of motion. The two components can then be resolved as a
right handed or left handed spinning field. Similarly the QM photon has either
linear polarization or circular polarization. For a photon represented as a
spin 1 helicity state |1,1>, |1,-1> the field rotates occupying all
orientation in the two dimensional plane.Notice that having a component in the x direction merely specifies that
the amplitude of a sine wave at its maximum value and that the field oscillates
through + to – orientation. Two photons with 180o phase difference
but the same polarization can cancel so that the overall Electric field
(polarization) is zero.Two x component
photons therefore can combine to provide 0 spin and two y component photons can
combine to provide 0 spin. [See http://beige.ovpit.indiana.edu/B679/node70.htmlfor a more extended derivation of the
relationship between linear and circular.]

Authors describing the two-photon systems often choose
linear polarization states. An apparatus similar to the Stern-Gerlach type
filter that Feynman describes can be made for photons. Although magnetic fields
cannot be used to split the photon into its two basis states, there are mirrors
that transmit one polarization but reflect the opposite polarization (beam
splitter cubes often reflect and transmit orthogonal polarization). One could
separate and then recombine using optical elements. Whatever the mechanism we imagine
that photon 1 enters the apparatus, splits based on polarization, travels both
paths (A and B) and reassembles to the identical QM state at the output. If a
path is blocked (e.g. the y component path) then the emerging particle is
polarized 100% in the x direction. Not all photons will emerge from a blocked
filter but some will.

For the two photon state, we can represent the the |0,0>
overall spin 0 state as:

Spin 0 is always a symmetric combination of the two
polarization states no matter what coordinate axes are chosen (x-y or x’y’).

Now if the |Ξ> state finds a filter (system 3 with
path A and B) it will add an additional component to the wf |3,0>. The
system for transporting the photon along two paths is labeled as 3 and its
state is a neutral state labeled 0.Assume that the original system |X1> is written with respect to a
filter oriented at an angle θ.

Since the photon travels through system 3 without changing
the state it contributes just a common factor and does not become entangled in
the quantum state. Of the four terms (1,3) travel along path A and (2,4) along
B. If path A is blocked.

The photon2 state must be in the same polarization at photon
1.The probability for observing this
is of course ½. An arbitrary polarization state for photon 1 will be measured
to be |1Y’> half of the time. When you observe the photon leaving you can
say with 100% confidence that photon 2 is also polarized in this direction
|2Y’>. Let us look at this problem a different way. Imagine that there is a
small amount of gas along path A and that the photon with an amplitude α
to Compton or Raleigh scatter so that system 3 changes. The state now becomes
|3,1> rather than |3,0>. This is an interesting case because it is
similar to slit electron diffraction with a light source to observe the slit.
It also perhaps is easier to see that the photon that travels down A is still
part of the system but that it is now in an orthogonal state due to the
presence of a scattered part of system 3. The above example illustrated the
effect of the orientation of the filter with respect to the initial choice of
basis. It would simplify the problem if we simply chose to express the state
with respect to that orientation.

Note these final states are entangled and one can’t factor
out system 3. This forces the quantum states to be labeled by the presence of a
changed system 3. One can consider the case α=1, β=0. This is the
same as blocking the (a) path since the final state always contains a new state
|3,1> and is therefore orthogonal.

In this situation if you measure the result of scattering
|3,1> you get one polarization state for photon 2 and if there is no change
in the measurement apparatus (system 3) you get another.

When considering the implications of quantum mechanics it is
sometime confusing to sort and arrange the various ideas. Critical to the
understanding is the assertion that composite systems behave by multiplying the
states of each system as shown above.When the new composite states are formed they cannot always be written
as a separated product of two isolated systems.The entangled states are those that can’t be written in the
form|1,A> |2,α> where all
the pieces of the 1 state can be collected together and multiplied by all the
pieces of the 2 state.The two-photon
state shown above is, of course, an example of entanglement.The QM predictions for these states are
somewhat counterintuitive. A measurement of photon 1 somehow determines the
state of photon 2. Without the measurement tests on photon 2 would conclude
that the photon was in an arbitrary state of polarization. Any Polaroid filter
would see a 50-50 split at all orientations. The QM wave function used to
describe photon 2 would be 1/√2 (|2X> + |2Y>) as far as observers
could tell. Somehow however should observers of photon 1 ever compare notes
with observers of photon 2 they would find 100% correlation between the
photons.